A classification of SU(d)-type C*-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category C with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from C such that there exists a *-representation of the Hecke algebra H-n(q) into C. The categorical trace on C induces a Markov trace on H-n(q). Using this Markov trace and a representation of H-n(q) in Rep(SU (root q)(d)) we show that C is equivalent to a twist of the category Rep(SU (root q)(d)). Furthermore a sufficient condition on a C*-tensor category C is given for existence of an embedding of a twist of Rep(SU (root q)(d)) in C.